Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682638 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #61337 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1595 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,9,11,6)(4,12,13,7)(10,14,16,15)(17,18,22,25)(19,20,26,23)(21,24,30,32)(27,28,31,29);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,18)(19,20)(21,27)(22,25)(23,26)(24,29)(28,32)(30,31);(17,19,23,18,22,26,20,25)(21,27,29,24,30,31,28,32);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,20,22,23)(18,19,25,26)(21,29,30,28)(24,31,32,27);(3,11)(6,9)(10,16)(14,15)(17,21)(18,24)(19,27)(20,28)(22,30)(23,29)(25,32)(26,31)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591352);